Good intuitions. FCC regulations restrict EIRP[0], which is essentially "what is the signal strength in the strongest direction". It's possible that increasing directionality of your antenna puts you over the EIRP limit (given how routers already try to get _really_ close to it to improve performance).
Basically APs on 2.4 GHz are only allowed 36 dBm (4 W) EIRP, no matter what antenna shape. This is usually not a problem though, since stations tend to have weaker transmitters anyway, but will still benefit from the AP's enhanced antenna gain.
The reason this is important is that it prevents private corporations from interfering with public air space (as discussed previously on HN [1]), and prevents landlords from interfering with the rights of tenants to legally place outdoor antennas (such as satellite dishes).
RF energy doesn't quite work like this. If your transmitter has a high EIRP (either due to antenna directionality or high TX power), then its signal is going to travel a long way, even after the signal level has dropped to that of a lower-EIRP transmitter. That is simply the inverse square law at work.
You could place sufficiently absorptive materials that this isn't so (e.g. enclose your house in a Faraday cage). But as hydrogen18 points out you're still limited by the FCC.
I think FCC laws technically apply within your property as well. Hypothetically, what happens if your non-compliant device blocks a visitor on your property from calling 911?
The FCC has stated that the RP-SMA connector (the connector used on nearly all wifi routers that have detachable antennas) shouldn't be allowed on part 15 devices, but also that it will delay enforcement of this indefinitely[1]. So even the FCC is fairly "meh" on enforcing this unless actual interference is reported.
Citation? Plenty of commercial part 15 devices don't even come with antennas, requiring you to "mix and match", unless I misunderstood what you meant by that.
4 W (36 dBm), and the 8.5 dBi gain of this homemade Yagi puts you over that if you're operating at 1 W (30 dBm) max transmit power (as some home routers do).
FCC regulates EIRP, which is output+antenna gain. For this to be strictly legal you would need to adjust the power down until the EIRP is within the legal limit. Actually to be strictly legal I think it would need to be type excepted by the FCC.
That being said if you're using this in your home/office on a legal channel it would be really difficult to even tell from outside you were using it and almost certainly not worth the FCCs time to even try to watch for it. It's not that much extra gain over the stock antenna.
To be fair, the FCC regulates both EIRP (4 W / 36 dBm) and max transmit power (1 W / 30 dBm). E.g. you can't just have a 34 dBm transmitter hooked up to a 2 dBi dipole.
If it's like here in Australia, they do consider EIRP. So a directional antenna would increase the EIRP, but with just 100mw tx at the output, I doubt EIRP would be over the limit (4W for 2.4Ghz ISM in Australia).
No, because no-one wants a home AP that only works if it's pointed at you.
You can buy directional APs if you want, but they're usually used for outdoor point-to-point links, or as sector antennas. e.g. I just installed one of these today: https://routerboard.com/RBSXTG-2HnD in a place that needed a signal far from where the AP was. Usually patch antennas (like this product) are preferred over Yagis for microwave links due to their smaller footprint, and capability of dual polarization in a single package.
Better is if your indoor AP can transceive in more of a "pancake" shape, like this: https://routerboard.com/RBOmniTikU-5HnD Note that's almost the same gain as this home-grown Yagi (7.5 dBi vs. 8.5 dBi) but with a more useful directionality.
Antenna gain is a function of directionality and vice versa. Patch antennas are also good because they have often very low "backward gain" but at the same time they are not very directional forward, so they are good to be placed in a corner of a room or near a wall.
It's not illegal per se, if you adjust transmission power accordingly. Router manufacturers generally don't use yagi because it's directional, requiring aiming, unlike omnidirectional dipole antenna. Modern wifi stations have much more sophisticated ways to increase range than passive antenna designs.
It's only illegal if you get net gain out of it[1]; if you turn down the power so that the maximum power transmitted in any direction is the same as before, then it's legal. You can get a small performance improvement by using directional antennas in this way as you reduce the received strength of off-axis signals which can reduce both multipath interference as well as interference from other stations.
It's not all that useful indoors though as 2.4 and 5GHz signals bounce off of walls really well so you lose much the directionality at the very first wall it hits.
1: Actually there is a total ERP requirement, but most routers with external antennas are already close to that requirement, so more than about 3dB gain will often put you over.
[0] https://en.wikipedia.org/wiki/Equivalent_isotropically_radia...